For $ E\subset \mathbb{ F}_q^d$, let $\Delta(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{ F}_q^d $ are subsets with $|E||F|\gg q^{ d+\frac{ 1}{ 3}}$ then $|\Delta(E)+\Delta(F)|> q/2$. They also proved that the threshold $q^{ d+\frac{ 1}{ 3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal $L^2$ restriction theorem for the sphere of zero radius.
Categories: math.NT math.CA